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### Alternating Sequence

```
Date: 01/27/97 at 20:43:09
From: KEITH MCKENZIE
Subject: I need help

Dear Dr. Math,

I need help. I have a problem that neither my friends nor I can get.
I thought you could help.  We have to find a pattern and find the next
three numbers for this: 0, 8, 27, _, _, _.  I hope you can find the

Sincerely,
Shoushou
```

```
Date: 01/29/97 at 16:53:47
From: Doctor Wallace
Subject: Re: I need help

Dear Shoushou,

For this series, the first thing to notice is that all of the numbers
are perfect cubes:  0 = 0^3, 8 = 2^3, and 27 = 3^3.

If we list the numbers that are the cube roots of these numbers, we
get this series:

0, 2, 3...

Now, all you have to do is find a pattern among these three.  The one
I see is an alternating series.  There may be others.  Let me know how
your work goes on this problem.

-Doctor Wallace,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 01/29/97 at 18:49:05
From: KEITH MCKENZIE
Subject: Re: I need help

This is Shoushou.  Thanks for giving me that great hint.   I'm not
sure if I have it right, but I think it might be +2, +1, +2, +1.  This
doesn't quite satisfy me, but it's the best I can come up with for
now.

Sincerely,
Shoushou
```

```
Date: 01/30/97 at 12:37:59
From: Doctor Wallace
Subject: Re: I need help

Dear Shoushou,

I think you're right.  That's what I came up with, too.  And I feel
the same way - I wasn't quite satisfied.  Somehow, it didn't feel
just right.  I think it was because there were only 3 numbers given,
and it's hard to come to just one pattern with only three numbers. But
it seems that, since the number 1 was missing, +2 +1 +2 +1 ... is the
only thing that makes sense.

Then the next two numbers in the pattern would be 5^3 and 6^3.  What
did your teacher have in mind for the answer?  If you can come up with
another pattern that fits the series, please write and let me know.

And write again any time you have other math questions!

-Doctor Wallace,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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